Experimental realization of quantum nondemolition (back-action-evading) measurement of a harmonic oscillator with sensitivity below the standard quantum limit for weak-force detection

The 1978 proposal by Thorne, Drever, Caves, Zimmermann, and Sandberg introduced quantum nondemolition (QND) measurement of a single quadrature of a harmonic oscillator — continuous, precise monitoring of a constant-of-motion observable with a linear coupling — as the route around the standard quantum limit (SQL) that constrains conventional readouts of weak-force detectors such as gravitational-wave antennas. The paper gave concrete schemes (a continuous LC-circuit/torsion-pendulum transducer requiring strong coupling; stroboscopic pulsed measurements vulnerable to timing jitter; and a near-nondemolition modulated readout whose back-action force must have negligible Fourier components at the modulation sideband frequencies) but no experimental realization. Single-quadrature back-action-evading measurements are now demonstrated in mesoscopic optomechanical, electromechanical, and spin systems, with sub-zero-point imprecision and several dB of conditional squeezing; and the LIGO detectors, via frequency-dependent squeezing, have operated with quantum noise up to 3 dB below the SQL since 2023-2024. What remains unrealized is the conjunction for which the schemes were conceived: a strict-criterion nondemolition (back-action-evading) readout of a conserved mechanical observable — a single quadrature $X_1$ of a harmonic oscillator or, in the free-mass limit, its conserved momentum or speed — with measurement sensitivity below the standard quantum limit for detecting weak classical forces in the gravitational-wave-relevant regime. The question is open.

Background

The weakest forces accessible to laboratory physics — most prominently gravitational waves (GWs) — are sought by monitoring the motion of high-quality harmonic oscillators: resonant bars in the 1970s, and today the kilogram-scale test masses of laser interferometers such as LIGO. A harmonic oscillator with generalized coordinate $x$, momentum $p$, mass $m$, and angular frequency $\omega$ is equivalently described by two slowly varying quadratures, defined through $X_1+iX_2=(x+ip/m\omega)e^{i\omega t}$. In the absence of driving, $X_1$ and $X_2$ are constants of the motion, and $(m\omega)^{1/2}X_1$ and $(m\omega)^{1/2}X_2$ are canonically conjugate observables: $[X_1,X_2]=i\hbar/m\omega$. A conventional readout monitors $x(t)$ over one or more cycles and therefore attempts to determine both quadratures simultaneously with comparable precision, which the uncertainty principle bounds: $\Delta X_1=\Delta X_2\gtrsim(\hbar/2m\omega)^{1/2}$, equivalently $\Delta N\gtrsim(N+1/4)^{1/2}$ for the phonon number $N$ (the number of quanta of excitation). This is the standard quantum limit (SQL), first derived by Braginsky for optical-lever and Fabry-Perot (resonant optical cavity) sensors and by Giffard for any linear readout; the 1978 source paper generalizes it to any scheme that measures both quadratures. Gravitational waves from supernovae at the Virgo cluster would drive detectors of reasonable mass below this limit, so the SQL was recognized early as the fundamental obstacle to GW detection.

A quantum nondemolition (QND) measurement is one that can be repeated on the same system without perturbing the measured observable: the back-action of the meter — the random force that any measurement necessarily exerts, e.g., quantum radiation-pressure noise from the photons used to sense position — is shunted entirely into the conjugate observable (back-action evasion, BAE). Braginsky's school had proposed measuring the phonon number $N$, but that requires a coupling quadratic in the oscillator variables, very hard to realize for weak signals, and a sequence of $N$ measurements reveals only the spectral density of the driving force near resonance, not its waveform. Thorne, Drever, Caves, Zimmermann, and Sandberg instead proposed measuring $X_1$ alone, with a linear interaction Hamiltonian $H_I=K\hat X_1\hat\varphi$ ($\hat\varphi$ a meter variable, $K$ a coupling constant); since $dX_1/dt=0$, the measurement is nondemolition, and the force follows from the measured eigenvalue $\xi(t)$ via $F(t)\sin\omega t=-m\omega\, d\xi/dt$. Concrete schemes were given: (i) a continuous transducer coupling an electrical LC oscillator (inductance $L$, capacitance $C$) through charge and current sensors to a torsion pendulum read out by a torque balance, with imprecision $\Delta X_1\simeq(\hbar/L\omega)^{1/2}(I\omega^2/K^2C)^{1/2}(\omega\tau)^{-3/2}$ for pendulum moment of inertia $I$ and averaging time $\tau$ — beating the SQL with sub-cycle averaging only in the strong-coupling limit $K^2C/(I\omega^2)\gtrsim1$, which the authors judged "may be very hard to achieve in practice"; (ii) stroboscopic (pulsed) measurements, coupling to $x$ only at phases $\omega t=n\pi$ where $x=(-1)^nX_1$, with timing errors $\Delta t$ (jitter or finite pulse duration) producing a spread $\Delta X_1\sim[(\hbar/m\omega)(\omega\Delta t)]^{1/2}$; (iii) a near-nondemolition variant, coupling a coordinate sensor sinusoidally ($H_I=K\hat\varphi\hat x\cos\omega t$) and averaging over many cycles ($\tau\gg2\pi/\omega$), which requires the readout's back-action force to have negligible Fourier components at the modulation sideband frequency — stated as $2\omega$ in the companion review (Caves et al., Rev. Mod. Phys. 52, 341 (1980), Sec. II.F). Extensions to mechanical oscillators and microwave cavities were sketched in one paragraph. None of this was demonstrated; the paper's own outlook flags the strong-coupling requirement, timing jitter, and the untested near-nondemolition condition as the practical obstacles.

The subsequent program unfolded in three lines. Kimble, Levin, Matsko, Thorne, and Vyatchanin translated QND ideas into interferometric GW detectors ("quantum nondemolition interferometers"), where photon shot noise (measurement imprecision from photon-counting fluctuations) and quantum radiation-pressure noise (back-action) set the free-mass SQL; their toolkit — squeezed-light injection (light with noise in one optical quadrature reduced below the coherent-state level), variational readout (measuring an optimized output light quadrature so that back-action is evaded), and speed meters (topologies that measure the conserved momentum of the test masses; practical designs by Purdue and Chen, prototype by the Glasgow Sagnac speed-meter experiment) — is the modern form of the 1978 proposal. Clerk, Marquardt, and Jacobs recast two-tone and stroboscopic BAE for cavity optomechanics (the coupling of optical cavity fields to mechanical motion via radiation pressure). Experimentally, single-quadrature BAE of a mechanical oscillator was first realized in a superconducting electromechanical device by Hertzberg et al.; Suh et al. achieved imprecision below the zero-point fluctuations (the ground-state motion amplitude $(\hbar/2m\omega)^{1/2}$) with 9 dB back-action evasion; Vasilakis et al. realized the stroboscopic scheme in a collective-spin oscillator (an atomic ensemble whose collective spin components play the role of the oscillator quadratures) with 2.2 dB of conditional squeezing (noise below the zero-point level on the quadrature inferred from the measurement record); Lei et al. used an independent BAE measurement to quantify a 4.7 dB-squeezed mechanical state; Shomroni et al. realized two-tone BAE in the optical domain. In parallel, LIGO implemented frequency-dependent squeezing (squeezed vacuum reflected from a 300 m filter cavity, which rotates the squeezed quadrature with frequency so that shot noise and radiation-pressure noise are both reduced) and in 2024 reported detector quantum noise up to 3 dB below the SQL — by quantum-noise engineering rather than by a nondemolition readout of the test masses.

Problem Statement

Can the single-quadrature quantum nondemolition (back-action-evading) measurement proposed by Thorne, Drever, Caves, Zimmermann, and Sandberg (1978) be experimentally realized — precise, repeatable monitoring of a single conserved quadrature $X_1$ of a harmonic oscillator (or, in the free-mass limit $\omega\to0$, of the conserved momentum or speed) through a linear interaction of the form $H_I=K\hat X_1\hat\varphi$ ($\hat\varphi$ a meter variable, $K$ a coupling constant) — with measurement sensitivity below the standard quantum limit for detecting weak classical forces in the gravitational-wave-relevant regime for which it was conceived? "The gravitational-wave-relevant regime" here means operation in the 10 Hz-to-10 kHz detection band of the resonant-bar and laser-interferometric detectors for which the schemes were conceived and at a displacement (equivalently force) sensitivity at the gravitational-wave-antenna level — for a kilogram-scale test mass, of order $10^{-20}\,\mathrm{m}/\sqrt{\mathrm{Hz}}$. Here "nondemolition" is used in the strict sense of the source lineage (Thorne et al. 1978; Caves et al., Rev. Mod. Phys. 52, 341 (1980), Sec. IV.A): the measured observable $A$ is a continuous QND observable — a sequence of precise measurements of $A$ whose results (after the first) are completely predictable, in the absence of a classical force, from the result of the preceding measurement, equivalently $A(t)=f[A(t_0)]$ and $[A(t),A(t')]=0$, as holds for conserved quantities such as $X_1$ or free-mass momentum — and the measurement back-action is deflected into the conjugate observable. "Below the standard quantum limit" refers to the bound-oscillator quadrature SQL $\Delta X_1=\Delta X_2\gtrsim(\hbar/2m\omega)^{1/2}$ and, for continuous readout of a free test mass, to the free-mass SQL of the companion review (Caves et al. 1980, Sec. III.A). Correlation-based, SQL-beating quantum-noise engineering — such as the frequency-dependent squeezing with which the LIGO detectors reduced detector quantum noise by up to 3 dB below the SQL — is termed "QND" in parts of the gravitational-wave literature (Braginsky's device-level naming; Kimble et al. 2002 list the squeezed-input interferometer among their "QND interferometer" designs), but it continues to measure the non-conserved mirror position rather than a continuous QND observable of the test masses, and therefore does not by itself constitute such a nondemolition measurement or resolve the question.

The single scientific objective is a working weak-force readout that measures a conserved observable in the strict sense above and whose added noise (imprecision plus residual back-action) on that observable is below the standard quantum limit, at weak-force detection sensitivity in the gravitational-wave-relevant regime for which the schemes were conceived. The source's example measuring systems are alternatives, each carrying scheme-specific obstacles that a realization must address for whichever scheme it uses: (1) for continuous-transducer schemes, achieving single-quadrature coupling strong enough that sub-SQL precision is reached at averaging times short compared with the signal variation — for the LC-circuit/torsion-pendulum transducer, the strong-coupling condition $K^2C/(I\omega^2)\gtrsim1$ (with $K$ the coupling constant, $C$ the transducer capacitance, $I$ the pendulum moment of inertia, and $\omega$ the oscillator angular frequency); (2) for stroboscopic (pulsed) schemes, holding timing jitter and pulse duration below the level at which the induced spread $\Delta X_1\sim[(\hbar/m\omega)(\omega\Delta t)]^{1/2}$ dominates the target noise; and (3) for the modulated near-nondemolition readout, realizing a back-action force spectral density negligible at the modulation sideband frequencies, together with the required quantum-limited transducers and amplifiers. Existing demonstrations do not by themselves answer the question: the mesoscopic single-quadrature back-action-evading measurements realize the measurement principle but at force sensitivities far from the gravitational-wave-relevant regime, and the sub-SQL operation of LIGO via frequency-dependent squeezing measures a non-conserved position observable rather than performing a nondemolition measurement of a conserved mechanical observable.

References

1. Kip S. Thorne, Ronald W. P. Drever, Carlton M. Caves, Mark Zimmermann, Vernon D. Sandberg (1978). Quantum Nondemolition Measurements of Harmonic Oscillators. Physical Review Letters 40, 667-671. DOI: 10.1103/PhysRevLett.40.667. https://doi.org/10.1103/PhysRevLett.40.667 (source of the proposal)
2. Carlton M. Caves, Kip S. Thorne, Ronald W. P. Drever, Vernon D. Sandberg, Mark Zimmermann (1980). On the measurement of a weak classical force coupled to a quantum-mechanical oscillator. I. Issues of principle. Reviews of Modern Physics 52, 341. DOI: 10.1103/RevModPhys.52.341. https://doi.org/10.1103/RevModPhys.52.341 (full text: https://authors.library.caltech.edu/records/1zzhw-ypw31/files/CAVrmp80.pdf?download=1; Sec. IV.A states the strict QND-observable criterion attributed to Thorne et al. 1978, Secs. II.F-III.A the scheme-specific limits and the free-mass SQL)
3. Vladimir B. Braginsky, Yuri I. Vorontsov, Kip S. Thorne (1980). Quantum Nondemolition Measurements. Science 209, 547. DOI: 10.1126/science.209.4456.547. https://doi.org/10.1126/science.209.4456.547
4. R. P. Giffard (1976). Ultimate sensitivity limit of a resonant gravitational wave antenna using a linear motion detector. Physical Review D 14, 2478-2486. DOI: 10.1103/PhysRevD.14.2478. https://doi.org/10.1103/PhysRevD.14.2478
5. Mark F. Bocko, Roberto Onofrio (1996). On the measurement of a weak classical force coupled to a harmonic oscillator: experimental progress. Reviews of Modern Physics 68, 755. DOI: 10.1103/RevModPhys.68.755. https://doi.org/10.1103/RevModPhys.68.755
6. E. Majorana, N. Pergola, P. Puppo, P. Rapagnani, F. Ricci (1993). Observation of the Brownian motion of a mechanical oscillator by means of a back action evading system. Physics Letters A 180, 43-49. DOI: 10.1016/0375-9601(93)90492-I. https://doi.org/10.1016/0375-9601(93)90492-I
7. H. J. Kimble, Yuri Levin, Andrey B. Matsko, Kip S. Thorne, Sergey P. Vyatchanin (2002). Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics. Physical Review D 65, 022002. DOI: 10.1103/PhysRevD.65.022002. https://doi.org/10.1103/PhysRevD.65.022002 (preprint arXiv:gr-qc/0008026, https://arxiv.org/abs/gr-qc/0008026)
8. Patricia Purdue, Yanbei Chen (2002). Practical speed meter designs for quantum nondemolition gravitational-wave interferometers. Physical Review D 66, 122004. DOI: 10.1103/PhysRevD.66.122004. https://doi.org/10.1103/PhysRevD.66.122004
9. A. A. Clerk, F. Marquardt, K. Jacobs (2008). Back-action evasion and squeezing of a mechanical resonator using a cavity detector. New Journal of Physics 10, 095010. DOI: 10.1088/1367-2630/10/9/095010. https://doi.org/10.1088/1367-2630/10/9/095010
10. J. B. Hertzberg, T. Rocheleau, T. Ndukum, M. Savva, A. A. Clerk, K. C. Schwab (2010). Back-action-evading measurements of nanomechanical motion. Nature Physics 6, 213-217. DOI: 10.1038/nphys1479. https://doi.org/10.1038/nphys1479 (preprint arXiv:0906.0967, https://arxiv.org/abs/0906.0967)
11. J. Suh, A. J. Weinstein, C. U. Lei, E. E. Wollman, S. K. Steinke, P. Meystre, A. A. Clerk, K. C. Schwab (2014). Mechanically detecting and avoiding the quantum fluctuations of a microwave field. Science 344, 1262-1265. DOI: 10.1126/science.1253258. https://doi.org/10.1126/science.1253258 (preprint arXiv:1312.4084, https://arxiv.org/abs/1312.4084)
12. Georgios Vasilakis, Heng Shen, Kasper Jensen, Misha Balabas, Daniel Salart, Bing Chen, Eugene S. Polzik (2015). Generation of a squeezed state of an oscillator by stroboscopic back-action-evading measurement. Nature Physics 11, 389-392. DOI: 10.1038/nphys3280. https://doi.org/10.1038/nphys3280 (preprint arXiv:1411.6289, https://arxiv.org/abs/1411.6289)
13. C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, K. C. Schwab (2016). Quantum Nondemolition Measurement of a Quantum Squeezed State Beyond the 3 dB Limit. Physical Review Letters 117, 100801. DOI: 10.1103/PhysRevLett.117.100801. https://doi.org/10.1103/PhysRevLett.117.100801 (preprint arXiv:1605.08148, https://arxiv.org/abs/1605.08148, posted under the differing preprint title 'Quantum nondemolition measurement of mechanical squeezed state beyond the 3 dB limit')
14. Itay Shomroni, Liu Qiu, Daniel Malz, Andreas Nunnenkamp, Tobias J. Kippenberg (2019). Optical backaction-evading measurement of a mechanical oscillator. Nature Communications 10, 2086. DOI: 10.1038/s41467-019-10024-3. https://doi.org/10.1038/s41467-019-10024-3 (preprint arXiv:1809.01007, https://arxiv.org/abs/1809.01007)
15. C. Gräf, B. W. Barr, A. S. Bell, F. Campbell, A. V. Cumming, S. L. Danilishin, N. A. Gordon, G. D. Hammond, J. Hennig, E. A. Houston, et al. (2014). Design of a speed meter interferometer proof-of-principle experiment. Classical and Quantum Gravity 31, 215009. DOI: 10.1088/0264-9381/31/21/215009. https://doi.org/10.1088/0264-9381/31/21/215009
16. S. S. Leavey, S. L. Danilishin, A. Gläfke, B. W. Barr, A. S. Bell, C. Gräf, J.-S. Hennig, E. A. Houston, S. H. Huttner, H. Lück, et al. (2016). Control of a velocity-sensitive audio-band quantum non-demolition interferometer. arXiv:1603.07756. https://arxiv.org/abs/1603.07756
17. D. Ganapathy, W. Jia, M. Nakano, V. Xu, N. Aritomi, et al. (LIGO O4 Detector Collaboration) (2023). Broadband Quantum Enhancement of the LIGO Detectors with Frequency-Dependent Squeezing. Physical Review X 13, 041021. DOI: 10.1103/PhysRevX.13.041021. https://doi.org/10.1103/PhysRevX.13.041021
18. W. Jia, V. Xu, K. Kuns, M. Nakano, L. Barsotti, M. Evans, N. Mavalvala, et al. (LIGO O4 Detector Collaboration) (2024). Squeezing the quantum noise of a gravitational-wave detector below the standard quantum limit. Science 385, 1318-1321. DOI: 10.1126/science.ado8069. https://doi.org/10.1126/science.ado8069 (preprint arXiv:2404.14569, https://arxiv.org/abs/2404.14569)
19. Yuki Inoue, Hsiang-Chieh Hsu, Hsiang-Yu Huang, M. Afif Ismail, Vivek Kumar, Miftahul Ma'arif, Avani Patel, Daiki Tanabe, Henry Tsz-King Wong, Ta-Chun Yu (2025). CHRONOS: Cryogenic sub-Hz cROss torsion bar detector with quantum NOn-demolition Speed meter. arXiv:2509.23172. https://arxiv.org/abs/2509.23172